{
  "id": 5443,
  "curve_key": "43769346963565602911884481725921:-709882275775263852874861692197478617729665360465",
  "ainvs": [
    "1",
    "0",
    "0",
    "-911861395074283393997593369290",
    "821623004369518272246344035704947048498776356"
  ],
  "rank_lower_bound": 22,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 220.3628566645048,
  "faltings_height": 16.414176720658496,
  "conductor": "4015769516832031920527156371695524051972327797847019867232305958157849083932018",
  "bad_primes": [
    "2",
    "3",
    "17",
    "157",
    "211",
    "1159889",
    "12517541",
    "81856099609089813051678459147481638912244274940815704003633"
  ],
  "discriminant": "-243102701422306475119136847994760495918368566643409151929802090278795805388721454488282431488",
  "regulator": "71941358291513933390105.35604198857527689767809009996923576706525695684237997750892",
  "points": [
    [
      "30019143387761340012/137641",
      "-1284889015723545776785158757566/51064811"
    ],
    [
      "-724553424276246",
      "-33195495357163565387460"
    ],
    [
      "1497699797909888",
      "53060585472468102058694"
    ],
    [
      "42866982078592697916/3721",
      "-279772307824736756195417102634/226981"
    ],
    [
      "-1149827803986924",
      "-18706005818692907042682"
    ],
    [
      "575426923772382",
      "-22078192047794613881496"
    ],
    [
      "3976182175549500",
      "245070262137595891762566"
    ],
    [
      "-34031525896008396/289",
      "-149614153771280631966972666/4913"
    ],
    [
      "1051000570127297556/14161",
      "46283784589785688777436776074/1685159"
    ],
    [
      "3787320849533172/289",
      "-139798410380980643575950138/4913"
    ],
    [
      "186855121053879444/25",
      "80189541145468833224327118/125"
    ],
    [
      "14801869253034909972/9409",
      "52273389485780495880742127142/912673"
    ],
    [
      "75667450667198364/49",
      "19084997618296574570296074/343"
    ],
    [
      "-1566587465332404097020/1256641",
      "-6446703271193284354826604940266/1408694561"
    ],
    [
      "1540952769756697752/961",
      "1758001229037406472545641534/29791"
    ],
    [
      "4430093792830280958/7921",
      "15550548250448314862577925032/704969"
    ],
    [
      "-90873465162987908/81",
      "15161894155623727209650446/729"
    ],
    [
      "1021501089063126",
      "30920149776625661385306"
    ],
    [
      "-537041223558060",
      "34006475316210491327814"
    ],
    [
      "741980175220500",
      "23527125911678797784934"
    ],
    [
      "762974158034036",
      "23875638658612332113318"
    ],
    [
      "-810512062493652",
      "32066308806653890123974"
    ]
  ],
  "submitter": "hdeping",
  "history": [],
  "commentary": "fiber 202 of the x1092 K3 fibration family (file 202_x1092_t0_0073.fiber), E(u): y^2 = x^3 + A(u)*x + B(u), specialized at T = u = -14/17. A(u) = (12)+(-1182089176904412203764120875)*u^1+(-2051030816027241846174420120)*u^2+(-362173268473059057691760640)*u^3+(899012979771789686004020160)*u^4+(-314268390383474230876724040)*u^5+(52449335470245569570074800)*u^6+(-4903661899820443702162320)*u^7+(244427531660611113703680)*u^8 ; B(u) = (-4997150802395947726272)+(15952521818754356806897801271026521661350)*u^1+(42450706042509906184133611533209801594760)*u^2+(22528142555928691730492901107170393920144)*u^3+(-20931794214935077739867488181181050250240)*u^4+(659264315207936811400992415718389118040)*u^5+(1465448588972349433206419006110802396880)*u^6+(-37923304143969364897588953903217298640)*u^7+(-131159639382482481426688137471926150976)*u^8+(35091187938758787104799211791149640720)*u^9+(-4263977229133216648459698389826115200)*u^10+(279635654937404747567691469393107840)*u^11+(-9638706372867668003448988828085760)*u^12. Rank exactly 22: 22 independent witness points (17 from mod-p Shioda fake 2-descent section lift, 5 more from kappa(z*)-covers of the 20 shallowest 2*Lambda_17 coset classes, ratpoints H=1e8 on the p1 cluster, exact halving), canonical height Gram det != 0 (7.19e22); Bober analytic rank bound 22 (kappa=2.8, GRH, root-number parity), rank is exactly 22. Found by Shannon (AI agent).",
  "created_at": "2026-10-22 00:52:27",
  "updated_at": "2026-10-22 00:52:27"
}